A square plate of edge $d$ and a circular disc of diameter $d$ are placed touching each other at the midpoint of an edge of the plate, as shown in the figure. Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.

Calculate the velocity of the centre of mass of the system of particles shown in the figure.

Two blocks of masses $10 \mathrm{kg}$ and $30 \mathrm{kg}$ are placed along a vertical line. The first block is raised through a height of $7 \mathrm{cm}$. By what distance should the second mass be moved to raise the centre of mass by $1 \mathrm{cm}$.

Consider a gravity-free hall in which a tray of mass $M,$ carrying a cubical block of ice of mass $m$ and edge $L,$ is at rest in the middle, as shown in the figure. If the ice melts, by what distance does the centre of mass of "the tray plus the ice" system descend?

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_1$ and $R_2$, as shown in figure.

A cart of mass $M$ is at rest on a frictionless horizontal surface and a pendulum bob of mass $m$ hangs from the roof of the cart. The string breaks, the bob falls on the floor, makes several collisions on the floor and finally lands up in a small slot made in the floor. The horizontal distance between the string and the slot is $L.$ Find the displacement of the cart during this process.

The balloon, the light rope and the monkey, shown in the figure, are at rest in the air. If the monkey reaches the top of the rope, by what distance does the balloon descend? Mass of the balloon $=M$, mass of the monkey $=m$ and the length of the rope ascended by the monkey $=L$.